Research Interests:
My primary research interest is geometric group theory, particularly studying infinite discrete groups through their actions on topological spaces. Thus far, my research has focused on automorphism groups of finitely generated free groups, Aut(Fn), and their actions on Outer space and Auter space. More recently, I have also been studying mapping class groups of surfaces via their actions on Ribbon Graph complexes.
Figure: The quotient of the degree two subcomplex of the spine of Auter space under the action of the automorphism group of the free group on four generators.
Publications:
To appear in Journal of Group Theory 11 (2008), 267-276.
We study the action of the Group Aut(Fn) of automorphisms of a finitely generated free group on the degree 2 subcomplex of the spine of Auter space. Hatcher and Vogtmann showed that this subcomplex is simply connected, and we use the method of K.S. Brown to deduce a new presentation of Aut(Fn).
Expository Talks:
I have been a regular participant in Cornell’s Berstein Seminar since Spring 2004, giving anywhere from two to seven talks each semester. In this seminar, graduate students give expository talks within a global topic that is set each semester. The topics are chosen mostly from topology and geometric group theory, though sometimes branch into combinatorics. The following is a list of talks/series of talks that I have given in the Berstein Seminar.
Spring 2008 - Length Spectra of Compact Constant Curvature Polyhedral Complexes
Spring 2007 - Length functions and actions on trees, the Culler-Morgan theorem
Fall 2006 - Gradient Vector Fields and Evasivness in Discrete Morse Theory
Fall 2006 - Homotopy Types of Subspace Arrangements (series given with Owen Baker and Gwyneth Whieldon)
Spring 2006 - Attacks on Braid Group Based Cryptosystems
Fall 2005 - An Application of Bass-Serre Theory to Aut(Fn), A Presentation for Aut(Fn) (series given with Heather Armstrong)
Fall 2005 - Existence of Uniform Tree Lattices
Spring 2005 - The Curve Complex of an Orientable Surface is Delta-Hyperbolic (series given with Nathan Broaddus)
Fall 2004 - Generating Aut(Fn) by Nielsen’s generators (series given with Heather Armstrong)
Fall 2004 - Isoperimetric Inequalities for Aut(Fn), and the Contractibility of Outer space
Spring 2004 - Generating sets for the Torelli Subgroup of the Mapping Class Group of an Orientable Surface
Research Talks:
15 March 2008
7 February 2008
Binghamton University
Geometry and Topology Seminar
22 January 2008
Middle Tennessee State University
Department of Mathematical Sciences Seminar
25 September 2007
Cornell University
Topology and Geometric Group Theory Seminar
Auter space, a cousin of Culler and Vogtmann's Outer space, was introduced by Hatcher and Vogtmann in 1998. In this introduction, they defined the degree subcomplexes of Auter space, which act like skeleta for Auter space. In this talk I will present several applications of the degree subcomplexes. Among these applications is a new presentation for Aut(Fn) which is joint work with Heather Armstrong and Karen Vogtmann. Also, I will discuss open problems that could potentially utilize the degree subcomplexes, including determining the Dehn function of Aut(Fn).
In the spirit of minimizing pre-requisites, I will present definitions for both Auter space and the degree subcomplexes of Auter space.